Modeling and Stability of PERT

Transcription

1 Modeling Stability of PET Yueping Zhang I. SYSTEM MODEL Our modeling of PET is composed of three parts: window adjustment ED emulation queuing behavior. We start with the window dynamics. Similar to [7] we consider a single-link scenario assume the forward propagation delay from the source to the router is negligible thus the round-trip (t measured by the end-user at t is composed of backward propagation delay T p queuing delay T q (t (t i.e. (t = T p + T q ( t (t. ( Denoting by C the link s capacity by q(t the queue size at t queuing delay T q (t can be approimated by q(t (t/c. ote that delay (t in the last epression is because the queuing delay perceived by the user at t is actually eperienced by the router (t units earlier. Then window dynamics of a PET end-flow is written as: Ẇ (t = (t W (tw ( t (t ( p(t ( t (t where at t W (t is the congestion window size (t is the TT p(t is the packet dropping probability. ote that loss rate p(t in the last equation is an instantaneous value as appose to its delayed counterpart p(t (t in the TCP/ED model obtained in [7]. This is because a PET user makes its dropping decision at the end-host instead of the router. To formulate PET s emulation of the ED mechanism assume that the propagation delay T p is known to the endflow (this can be approimated by the base TT. Then upon each packet arrival the user can estimate the queuing delay by T q (t = (t T p generate the packet drop probability p(t as following: p(t = T q(t T min T ma T min p ma (3 where T min T ma are the maimum minimum thresholds of queuing delays p ma is a constant. Another component of ED emulation is the estimation of round-trip (t which is updated per-packet using a lowpass filter (LPF with weight α i.e. (t = α(t + ( α ˆ(t ( where ˆ(t is the instantaneous TT measured at t weight α =.99. Following the technique used in [7] this LPF can be approimated by the following differential equation: Ṙ(t = ln α δ ((t ˆ(t (5 where δ is the sampling interval. We net model the queuing dynamics which can be described by the following differential equation of queue size: q(t = W (t (t C (t where (t is the number of flows accessing the router at t term W (t(t/(t can be interpreted as the combined incoming rate y(t. Since T q (t = q(t (t/c we re-write the last equation in terms of queuing delay T q (t: T W (t (t(t (t q (t. (6 (t (tc Then equations (-(6 comprise the complete system model of PET. It is easy to see that this model is very similar to the TCP/ED model derived in [7]. ote that there are also some differences between the system models of PET TCP/ED. The first one is that in PET loss rate p in ( is not delayed by (t variables for computing queuing delay T q in (6 are retarded by one TT. As we have mentioned earlier this difference is because PET monitors queuing dynamics at the end-user while ED does this inside the router. The second difference is that PET uses queuing delay T q (t to determine loss probability while ED uses queue length q(t. We net study stability of PET (-(6. A. Stability esult II. STABILITY AALYSIS Similar to [7] we assume that the number of flows TT are constant in the steady state. Then the system becomes: f(w W T q p = W (tw (t p(t g(w T q = W (t (7 C where W (t = W (t. In the steady state we have W (t = W (t = W. Applying this result equating the left-h sides of (7 to zero we can derive the equilibrium point (W p Tq as follows: W = C p = C. (8 The following theorem states a sufficient condition for PET (7 to be locally stable in its stationary point. Theorem : Let L P ET K be defined as follows: p ma L P ET = K = ln α T ma T min δ (9 assume bounds + satisfy the following condition: L P ET +3 C w g ( + ( K

2 where: ( w g =. min + C +. ( Then PET modeled by (7 is locally stable for all +. Proof: To determine local stability of the system we first linearize it in the equilibrium: δẇ (t = δw (t+ W δw (t+ W p δṗ(t = δw (t + δt q (t W T q δp(t+ T q δt q (t where δw (t = W (t W δw (t = W (t W δp(t = p(t p δt q (t = T q (t T q the partial derivatives are given below: W = W p W C ( W p = C (3 T q + W p = ( W = C (5 W T q C. (6 Using approimation W (t = W (t as in [7] the linearized PET model becomes: δẇ (t δ T q (t = C δw (t C δp(t (7 C δw (t δt q(t. (8 Taking Laplace transforms of both sides of the last two equations we obtain the following transfer functions: P W (s s + C C (9 P T = C s +. ( Further notice that according to [7] the transfer function of ED emulation (3-( in PET is: C(s = L P ET s/k +. ( Combining (9-( with the delay term e s in the return path we have the following closed-loop transfer-function model of PET: L(s = P W (sp T (sc(se s C e s L P ET (s + C (s + ( s K +. ( The rest of the proof directly follows from that of [ Proposition ]. Fig... sampling interval δ (sec 3 5 number of flows Sampling interval δ as a function of the minimum number of flows ote that in addition to conditions given in Theorem the equilibrium condition p p ma is also necessary for the system to be stable. To eamine validity of this condition we infer from (8 that p = /(W. Thus condition p p ma holds if only if p ma /(W. For a stationary window size W = packets we have p ma = % which is reasonable in practice. We net investigate whether there eists a closed-form solution of condition (-(. ecalling K = ln α/δ we convert ( into the following condition on δ: δ ln α ( L P ET w +6 C 6(. (3 g This equation can be used as the guideline for choosing sampling interval δ given C +. Moreover we observe that given + C δ scales inversely proportional to. To better see this consider the following simulation where T T = ms packet size s = 5 bytes p ma =. T ma = ms T min = 5 ms α =.99. We set C = mb/s (which corresponds to pkts/s range from to 5 to see the minimum requirement of δ. As seen from Figure the minimum δ monotonically decreases reaches. seconds as goes to. We note that is the lower bound of. Thus once a stable δ is picked given certain + C stability of the system is not affected by or as long as +. B. Matlab Simulations We net eamine this stability condition using Matlab simulations. Bringing in notations (t = (t = W (t (t = ˆ q (t 3 (t = q (t the fluid model of PET (-(6 can be transformed into the following delay differential equations (DDEs: ẋ (t = L P ET (t (t ( 3 (t T min ẋ (t = C (t ( ẋ 3 (t = K 3 (t K (t where K L P ET are defined in (9. Set link capacity C = pkt/s (or mb/s with packet size 5 bytes δ =. ms = = 5 p ma =.

3 (a = ms 3 5 (b = 6 ms 3 5 (c = 7 ms Fig.. The fluid model of PET (-(6 under different delay. T ma = ms T min = 5 ms α =.99. We keep = + test stability of ( under different values of delay. For all simulations we set the initial point to be ( the unit of (t is packets that of (t 3 (t are both seconds. Start with = ms which satisfies the stability condition in Theorem. As illustrated in Figure (a the system is stable with monotonic trajectories. We then increase to 6 ms which is closer to the stability boundary but still satisfies the stability condition. As shown in Figure (b the system is stable converges to its equilibrium (8 after decaying oscillations. Finally we increase to 7 ms which is eactly on the stability boundary. As seen from Figure (c the system is unstable ehibits persistent oscillations whose amplitude increases with the value of. ote that the stability boundary derived from Theorem is not eact. Even when is within the stability region but close enough to the boundary the system becomes unstable. This discrepancy is due to the approimations W (t = W (t (t in the proof. To justify this reasoning we rewrite ( using W (t = W (t (t repeat the above simulations. The simulation results demonstrate that stability of the system is guaranteed under condition ( becomes unstable when is increased to 75 ms (which eceeds the stability boundary 7 ms. Lack of necessity of the stability condition is also pointed out in the contet of TCP/ED in []. However Theorem still serves as a general guideline for choosing PET parameters. C. Discussion We remark on two differences between Theorem [7 Proposition ]. The first one is that in the left-h side of ( we have C instead of C 3 as in [7 (8]. This is because PET uses queuing delay T q (t to determine loss probability while ED uses queue size q(t. Applying L P ET = L ED C in ( it is evident that the resulting stability condition is identical to that of ED. The second difference is that sampling interval δ in ED is approimately fied to /C while in PET δ of user i is the inter-packet arrival / i (t which is approimately /C. As a consequence for fied C the more flows are accessing the bottleneck link the slower the sampling action of each flow. eflected in the stability condition this also results in less constraint on C + than in ED. Thus both differences actually increase the stability region of PET. On the other h when the frequency of sampling action becomes etremely low TT estimation at the end user may not accurately capture the queuing dynamics inside the router. However we argue that this problem does not occur in practice since the ISP will increase link capacity as the number of accessing flows becomes large to decrease packet loss rate prevent end-flows from starving. Thus there are always sufficient packets within one TT of each flow such that the end-user has enough samples for TT estimation. In addition it is common to assume that link capacity C scales linearly with the number of flows [] i.e. C/ = σ where σ is constant. Then assuming W = = + condition (-( translates into:. L P ET σ + σ K + (5 + which is independent of C is only a function of +. This property does not hold for ED due to term C 3 instead of C in (. As a consequence sampling interval δ of ED depends on ( thus unscalable to C even if C/ is kept constant. We should also note that similar to ED sampling interval δ in PET also depends on the link capacity C packet size s. Specifically δ becomes small when C increases or s decreases becomes large otherwise. As pointed out in [7] this adaptive nature of δ is harmful since small δ results in close tracking of the instantaneous queue length thus leads to large oscillations. On the other h large δ results in increased rise etended initial overshoot of the queue length. Thus it is suggested that δ be set to a fied value that is independent of C s. As mentioned above this requirement can be easily satisfied when ratio C/ is kept constant. A detailed discussion of the impact of δ on system performance is available in [7 Section 3.]. III. EMULATIG PI It has been identified that ED has several drawbacks such as lack of significant performance improvement for pure web traffic [] mitures of FTP UDP HTTP traffic [6] difficulties in tuning ED parameters [] [3] [7] tradeoff between stability responsiveness of the system [7]. As pointed out in [5] the major cause of the last drawback

4 in the above list is the averaging mechanism of the LPF. Thus the authors suggest using the instantaneous queue length applying a PI controller to it. Simulation results in [5] demonstrate that PI offers better performance than ED in terms of stability stationary queue size. Motivated by this fact we net seek to emulate PI inside PET. The transfer function of PI is: P (s C P I (s = T q (s = K + s/m (6 s where P (s T q (s are respectively the Laplace transforms of δp(t = p(t p δt q (t = T q (t Tq K m are constants to be determined net. Assuming p = we re-write PI (6 in the -domain as follows: ( δp(t = p(t = K δt q (t + δt q (tdt. (7 m To implement the continuous PI (7 in PET we discretize it using bilinear transform s = (z /δ(z+ where δ is the sampling interval convert the s-domain transfer function given in (6 into its following counterpart in the z-domain: C P I (z = P (z T q (z = γz β z (8 where γ = K/m + Kδ/ β = K/m Kδ/. Then the -domain version of the PI controller becomes: p(t = γ(t q (t T q β(t q (t T q + p(t (9 Thus in the new version of PET the only change is to replace ED emulation (3-( with the PI controller (9. Stability condition of PET/PI defined by (7 (9 is given below. Theorem : Assume: W (3 where W is the stationary window size. Then PET/PI defined by (7 (9 with m = + C K = m j m + (3 +3 C ( is locally stable for all flow numbers all stationary TT +. Furthermore the phase margin PM satisfies: P M 9 8 π m. (3 Proof: Following Theorem [5 Proposition ] it is easy to prove that PET/PI modeled by (7 (7 with K m defined in (3 is stable. Since stability is preserved under bilinear transform stability condition of PET (7 with discrete PI (9 is the same as that of (7 (7. Similar to Theorem since PET uses queuing delay to determine loss probability in parameter K (3 we have term C instead of C 3 as in the original stability condition of TCP/PI [5 Proposition ]. In addition it is clear that this controller is easy to implement does not require a complicated tuning process of parameters such as T min T ma p ma α as in ED. IV. PET-V In this section we present modeling stability analysis of a variation of PET which we call PET-V such that the resulting protocol can fairly compete with TCP SACK. The idea is to dynamically adjust the additive increase parameter α based on PET s early dropping probability p the link s packet loss probability p. ow the combined packet loss rate p P ET eperienced by the end-user becomes p P ET = ( p ( p = p+p pp. To roughly match the throughput equations of PET non-pet TCP flows we want to equate term βp P ET /α P ET βp/α. Since α = for the regular TCP we have α P ET = p P ET /p = + p /p p. Assuming p we simplify the last equation as α P ET = + p /p. We net analyze stability of the new method. Assuming loss rate p = p at links is constant we can rewrite steady-state equations of the system as following: ( + p /p W (p + p p p = W =. (33 C Then we have the following stationary values: W = C C p p p = C p + C. (3 p Local stability of PET-V is given in the following theorem. Theorem 3: Let L P ET K be defined as follows: L P ET = p ma T ma T min K = ln α δ (35 assume bounds + satisfy the following condition: L P ET + C( p ( ( ( + C p + ( + C p ( Cp w g + (36 K where: ( Cp w g =. min ( ( + C p + ( + C p + link loss rate p is assumed to be constant. Then PET modeled by (7 is locally stable for all +. Proof: Linearizing the system in its equilibrium we have: δẇ (t = δw (t+ δw (t+ W W p δp (t+ δt q (t T q δṗ (t = δw (t + δt q (t W T q where δw (t = W (t W δw (t = W (t W δp (t = p (t p δt q (t = T q (t T q the partial

5 5 derivatives are given below: W = W (p + p p p W Cp C p + C (37 p p W ( p C ( p (38 p + p T q p + W (p + p p p = (39 W = C ( W T q C. ( Using approimation W (t = W (t as in [7] the linearized PET model becomes: δẇ (t Cp C p + C δw (t p C ( p δp (t ( δ T q (t = C δw (t δt q(t. (3 Taking Laplace transforms of both sides of the last two equations we obtain the following transfer functions: P W (s s + P T (s = C ( p ( Cp C p + C p C s +. (5 Further notice that according to [7] the transfer function of ED emulation (3-( in PET is: C(s = L P ET s/k +. (6 Combining the above equations with the delay term e s in the return path we have the following closed-loop transferfunction model of PET: L(s = P W (sp T (sc(se s (7 (s + C( p e s L P ET Cp C p + C p (s + ( s K +. The rest of the proof directly follows from that of [ Proposition ]. A. Analysis V. PET-V3 In this section we present analyze a new version of PET which we call PET-V3. In this new protocol upon the onset of congestion the end-user reducing the congestion window by a factor of β = T q (t/(t q (t + T ma instead of β = /. In other words the congestion window W (t becomes ( βw (t upon detection of a packet loss event. Thus assuming the TT (t = is constant we can write the system equations as following: f(t = Ẇ (t = T maw (tw (t p(t (8 (T q (t + T ma g(t = q(t = W (t. (9 C Combining this model with equation p(t = L P ET (T q (t T min we have: W = C (5 T q = T ma( + C L P ET T min T ma C L P ET (5 p = L P ET (T ma + T min T ma C L P ET. (5 We net attempt to study local stability of the system in its equilibrium (W p Tq. Theorem : Let: L P ET = p ma T ma T min K = ln α δ. (53 Assume bounds + satisfy the following condition: (T ma + C L P ET w C L P ET T ma (T ma + T min g + (5 K where: ( w g =. min + C +. (55 Then PET-V3 modeled by (8-(9 is locally stable for all +. Proof: We first obtain the following partial derivatives: W = W T ma W (Tq + T ma p C (56 p p (57 = T q (Tq + T ma (58 W = C (59 T q. (6 Then the linearized window dynamics of PET-V3 is given by the following differential equation: δẇ (t δw (t C p δp(t + (Tq + T ma δt q(t. (6

6 (a = 5 ms 6 8 (b = 5 ms 6 8 (c = 7 ms Fig. 3. The fluid model of PET-V3 under different delay. Further noticing that T q (t = p(t/l P ET + T min we have T q = p /L P ET + T min δp(t = p(t p = δt q (t/l P ET. Then we can rewrite (6 as: where δẇ (t δw (t + Γδp(t (6 C (T ma C L P ET Γ C L P ET T ma (T ma + T min. (63 Then the s-domain transfer function of the last equation becomes: Γ P W (s. (6 s + C The linearized equation of queuing dynamics is the same as before: δ q(t = C δw (t δt q(t (65 whose transfer function is easy to obtain: P T (s = C s +. (66 Following the previous techniques we have the following closed-loop transfer function L(s: Γ L(s = C L P ET e s (s + C (s + ( s (67 K +. The rest of the proof directly follows from [ Proposition ]. We use the same system setting as Figure that is link capacity C = pkt/sec (or mb/s with packet size 5 bytes δ =. ms = = 5 p ma =. T ma = ms T min = 5 ms α =.99. We keep = + test stability of (68 under different values of delay. For all simulations we set the initial point to be ( the unit of (t is packets that of (t 3 (t are both seconds. We start with = 5 ms which satisfies the stability condition in Theorem. As illustrated in Figure 3(a PET-V3 is stable with monotonic trajectories converges the congestion window W (t eactly to the predicted value W = 3 packets. We then increase to 5 ms which is closer to the stability boundary but still satisfies the stability condition. As shown in Figure 3(b the system is stable converges to its equilibrium W = 5 packets after decaying oscillations. Finally we increase to the stability boundary 7 ms. As seen from Figure 3(c the system is marginally stable ehibits persistent oscillations. Comparing these results to Figure we infer that under the same system configuration PET-V3 ehibits better stability properties tolerates higher delay than the original PET. In addition we also observe from simulations that PET- V3 has lower stationary queuing delay Tq dropping rate p than PET as well as preserving the steady-state congestion window W. All of these properties combined its TCP friendliness make PET-V3 a better choice than PET in practical settings. B. Simulation We net eamine this stability condition using Matlab simulations. Bringing in notations (t = T p +T q (t (t = W (t (t = ˆT q (t 3 (t = T q (t the fluid model of PET-V3 can be transformed into the following delay differential equations: ẋ (t = ẋ (t = (t L P ET (tt ma ( 3 (t (t T min (t(t q (t + T ma (tc (t (68 ẋ 3 (t = K 3 (t K (t where K L P ET are defined in (9. EFEECES [] M. Christiansen K. Jeffay D. Ott F. D. Smith Tuning ED for Web Traffic in Proc. ACM SIGCOMM Aug. pp [] A. Dhamdhere H. Jiang C. Dovrolis Buffer Sizing for Congested Internet Links in Proc. IEEE IFOCOM Mar. 5 pp [3] S. Floyd. Gummadi S. Shenker Adaptive ED: An Algorithm for Increasing the obustness of ED s Active Queue Management ICI Tech. ep. Aug.. [] C. V. Hollot V. Misra D. Towsley W.-B. Gong A Control Theoretical Analysis of ED in Proc. IEEE IFOCOM Apr. pp [5] C. V. Hollot V. Misra D. Towsley W.-B. Gong On Designing Improved Controllers for AQM outers Supporting TCP Flows in Proc. IEEE IFOCOM Apr. pp [6] M. May J. Bolot C. Diot B. Lyles easons ot to Deploy ED in Proc. IEEE IWQoS Jun. 999 pp [7] V. Misra W.-B. Gong D. Towsley A Fluid-Based Analysis of a etwork of AQM outers Supporting TCP Flows with an Application to ED in Proc. ACM SIGCOMM Aug. pp. 5 6.